DiffEq21 - 6.2 Now that we have seen an example showing the significance of the eigenvalues and eigenvectors in understanding the behavior of the

DiffEq21 - 6.2 Now that we have seen an example showing the...

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6.2 The Behavior of 2 × 2 Linear Systems Now that we have seen an example showing the significance of the eigenvalues and eigenvectors in understanding the behavior of the solutions to a linear system of di ff erential equations, let’s systematically explore the 2 × 2 systems. Let A = a 1 , 1 a 1 , 2 a 2 , 1 a 2 , 2 . Then det ( A - λ I ) = det a 1 , 1 - λ a 1 , 2 a 2 , 1 a 2 , 2 - λ = λ 2 - ( a 1 , 1 + a 2 , 2 ) λ + ( a 1 , 1 a 2 , 2 - a 1 , 2 a 2 , 1 ) . (6) We call the sum of the diagonal entries of a matrix, M , the trace and denote it by tr( M ). Since tr( A ) = ( a 1 , 1 + a 2 , 2 ) and det( A ) = ( a 1 , 1 a 2 , 2 - a 1 , 2 a 2 , 1 ), we can write Equation 6 as det ( A - λ I ) = λ 2 - tr( A ) λ + det( A ) . (7) This means that to understand the eigenvalues of all 2 × 2 matrices we only need to consider two parameters, the trace and the determinant. Suppose that τ = tr( A ) and Δ = det( A ). Now let B = 0 - 1 Δ τ . Since tr( B ) = τ and det( B ) = Δ , A and B have the same eigenvalues. This means that to understand the eigenvalues for 2 × 2 matrices it is su ffi cient to only consider matrices of the form of B . Keep in mind that while A and B have the same eigenvalues, they will generally have di ff erent eigenvectors. However, it turns out that we can understand the qualitative behavior without knowing the eigenvectors. We should also note that the nullclines for the system d dt u ( t ) = Bu ( t ) = 0 - 1 Δ τ x ( t ) y ( t ) (8) are the x -axis and the straight line defined by y = - Δ x/ τ . Now suppose that the eigenvalues of a 2 × 2 matrix, A are λ 1 and λ 2 . Then we must have det ( A - λ I ) = ( λ - λ 1 )( λ - λ 2 ) = λ 2 - ( λ 1 + λ 2 ) λ + ( λ 1 λ 2 ) . (9) Thus tr( A ) = ( λ 1 + λ 2 ) and det( A ) = ( λ 1 λ 2 ) . This means that the trace of a 2 × 2 matrix is the sum of the eigenvalues, and its determinant is the product of the eigenvalues. It turns out that the same argument can be generalized to show that for any n × n matrix, the trace of the matrix is the sum of the eigenvalues and the determinant of the matrix is the product of the eigenvalues. 6
–2 –1 0 1 2 beta –3 –2 –1 1 2 3 alpha Figure 2: Trace-Determinant Diagram If we apply the quadratic formula to the quadratic λ 2 - τλ + Δ = 0 we get λ 1 = τ 2 + τ 2 2 - Δ λ 2 = τ 2 - τ 2 2 - Δ Armed with these relationships between the eigenvalues of a matrix and its trace and determinant, we can introduce the Trace-Determinant Diagram in Figure 2. We plot the trace,